Symplectic Geometry and Isomonodromic Deformations

Philip Paul Boalch
D.Phil. Thesis Submitted Trinity Term 1999

## Abstract

In this thesis we study the natural symplectic geometry of moduli spaces of meromorphic connections (with arbitrary order poles) over Riemann surfaces. The aim is to understand the symplectic geometry of the monodromy data of such connections, involving Stokes matrices. This is motivated by the appearance of Stokes matrices in the theory of Frobenius manifolds due to Dubrovin, and in the derivation of the isomonodromic deformation equations of Jimbo, Miwa and Ueno.

The main results of this thesis are:

• An extension to the meromorphic case of the infinite dimensional description, due to Atiyah and Bott, of the symplectic structure on moduli spaces of flat connections. This involves using an appropriate notion of singular $C^\infty$ connections and realises the natural moduli space of monodromy data as an infinite dimensional symplectic quotient.
• An explicit finite dimensional symplectic description of moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the Riemann sphere. A similar description is given of certain extended moduli spaces involving a compatible framing at each pole; these are the phase spaces of the isomonodromic deformation equations.
• A proof that the monodromy map is a symplectic map. In other words the above two symplectic structures are related by the transcendental map taking meromorphic connections to their monodromy data. The analogue of this result in inverse scattering theory is well-known and was important in developing the quantum inverse scattering method.
• A symplectic description of the full family of Jimbo-Miwa-Ueno isomonodromic deformation equations. In modern language we prove that the isomonodromic deformation equations are equivalent to a flat symplectic Ehresmann connection on a symplectic fibre bundle. This fits together, into a uniform framework, all the previous results for the six Painlevé equations and Schlesinger's equations.
• Finally we look at the simplest case involving Stokes matrices in detail. We present a conjecture relating Stokes matrices to Poisson-Lie groups (which we prove in the simplest case) and also prove directly that in low-dimensional cases the Poisson structure on the local moduli space of semisimple Frobenius manifolds does arise from a Poisson-Lie group.

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